We prove the generalized Hyers-Ulam stability of the wave equation, Δu = (1/c²)u_tt, in a class of twice continuously differentiable functions under some conditions.

1. Introduction

In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms.

Let G₁ be a group and let G₂ be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a function h : G₁ → G₂ satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G₁, then there exists a homomorphism H : G₁ → G₂ with d(h(x), H(x)) < ε for all x ∈ G₁?

The case of approximately additive functions was solved by Hyers [2] under the assumption that G₁ and G₂ are Banach spaces. Indeed, he proved that each solution of the inequality ∥f(x + y) − f(x) − f(y)∥ ≤ ε, for all x and y, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, f(x + y) = f(x) + f(y), is said to have the Hyers-Ulam stability.

Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows:

()

and proved the Hyers’ theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4–10].

The terminologies, the generalized Hyers-Ulam stability and the Hyers-Ulam stability, can also be applied to the case of other functional equations, of differential equations, and of various integral equations.

Given a real number c > 0, the partial differential equation

()

is called the wave equation, where u_tt(x, t) and Δu(x, t) denote the second time derivative and the Laplacian of u(x, t), respectively.

For an integer n ≥ 2, assume that U and T are open (connected) subsets of ℝⁿ and ℝ, respectively. Let φ : U × T → [0, ∞) be a function. If, for each twice continuously differentiable function u : U × T → ℝ satisfying

()

there exist a solution u₀ : U × T → ℝ of the wave equation (2) and a function Φ : U × T → [0, ∞) such that

()

where Φ(x, t) is independent of u(x, t) and u₀(x, t), then we say that the wave equation (2) has the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability).

In this paper, using ideas from [11, 12], we prove the generalized Hyers-Ulam stability of the wave equation (2).

2. Main Results

For a given integer n ≥ 2, x_i denotes the ith coordinate of any point x in ℝⁿ; that is x = (x₁, …, x_i, …, x_n), and |x| denotes the Euclidean distance between x and the origin; that is,

()

Given a real number c > 0, assume that real numbers a and t₂ satisfy a > c and 0 < t₂ < ∞, and define

()

We remark that (x, t) ∈ U × T if and only if |x | /t ∈ R. Using an idea from [11], we define a class W of all twice continuously differentiable functions u : U × T → ℝ with the properties

(i)
u(x, t) = tv(|x | /t) for all x ∈ U and t ∈ T and for some v : R → ℝ;
(ii)
.

If we define

()

for all u₁, u₂ ∈ W and λ ∈ ℝ, then W is a vector space over real numbers. That is, W is a large class such that it is a vector space.

Theorem 1. Let a function φ : U × T → [0, ∞) be given such that there exists a positive real number s with

()

If a u ∈ W satisfies the inequality

()

for all x ∈ U and t ∈ T, then there exists a solution u₀ : U × T → ℝ of the wave equation (2) which belongs to W and satisfies

()

for all x ∈ U and t ∈ T.

Proof. Let v : ℝ → ℝ be a function which satisfies

()

for all x ∈ U and t ∈ T. For any i ∈ {1,2, …, n}, we differentiate u(x, t) with respect to x_i to get

()

Similarly, we obtain the second partial derivative of u(x, t) with respect to x_i as follows:

()

Hence, we have

()

By a similar way, we further get the second derivative of u(x, t) with respect to t as follows:

()

Therefore, it follows from (14) and (15) that

()

for any x ∈ U, t ∈ T, and r∶ = |x | /t ∈ R, and it follows from (8) and (9) that

()

for all r ∈ R, where we set w(r)∶ = v^′(r).

Set

()

for each r ∈ R. Then we have

()

According to (18) and [13, Theorem 1], there exists a unique real number α such that

()

for all r ∈ R.

Hence, it follows from the last inequalities that

()

for any r ∈ R.

Due to (ii), it holds that . Replacing r with |x | /t in the last inequalities, we get

()

for all x ∈ U and t ∈ T.

If we define a function u₀ : U × T → ℝ by

()

then we have

()

for all x ∈ U and t ∈ T, which implies that u₀(x, t) is a solution of the wave equation (2).

It is now to show that u₀ ∈ W. Let F : R → ℝ be a function with the property

()

Then we have

()

which implies that u₀(x, t) can be expressed as tv(|x | /t), where v(r) = αF(r) − αF(a). Moreover, we get

()

which verifies that u₀ ∈ W. Finally, by (24), the inequality (10) holds true.

Assume now that b and t₁ are given real numbers satisfying 0 < b < c and 0 < t₁ < ∞. We then set

()

and define a class W^′ of all twice continuously differentiable functions u : U^′ × T^′ → ℝ with the properties

(iii)
u(x, t) = tv(|x|/t) for all x ∈ U^′ and t ∈ T^′ and for some v : R^′ → ℝ;
(iv)
.

It might be remarked that (x, t) ∈ U^′ × T^′ if and only if |x | /t ∈ R^′. If we define

()

for all u₁, u₂ ∈ W^′and λ ∈ ℝ, then W^′ is a vector space over real numbers.

Theorem 2. Let a function φ : U^′ × T^′ → [0, ∞) be given such that there exists a positive real number s^′ with

()

If a u ∈ W^′ satisfies the inequality

()

for all x ∈ U^′ and t ∈ T^′, then there exists a solution u₀ : U^′ × T^′ → ℝ of the wave equation (2) which belongs to W^′ and satisfies

()

for all x ∈ U^′ and t ∈ T^′.

Proof. If v : ℝ → ℝ is given by (11), then we can simply follow the lines in the first part of the proof of Theorem 1 to obtain

()

for all r ∈ R^′, where w(r)∶ = v^′(r).

Set

()

for any r ∈ R^′. Then we get

()

According to (35) and [13, Corollary 2], there exists a unique real number α such that

()

for all r ∈ R^′.

From the last inequalities, it follows that

()

for each r ∈ R^′.

On account of (iv), we have . Replacing r with |x | /t in the last inequalities, we obtain

()

for all x ∈ U^′and t ∈ T^′.

Let us define a function u₀ : U^′ × T^′ → ℝ by

()

Then, a similar argument to the last part of the proof of Theorem 1 shows that u₀(x, t) is a solution of the wave equation (2) and it belongs to W^′. Finally, the validity of (34) immediately follows from (41).

3. Remarks

Remark 1. The inequality (10) in Theorem 1 can be rewritten as

()

for all x ∈ U and t ∈ T. If we further substitute sinθ for c/z in the previous inequality, then we obtain

()

for any x ∈ U and t ∈ T.

For the case of n = 3, the inequality (10) can be rewritten as

()

for all x ∈ U and t ∈ T.

Remark 2. As in Remark 1, the inequality (34) in Theorem 2 can be rewritten as

()

for all x ∈ U^′ and t ∈ T^′. If we substitute c cos θ for z in the previous inequality, then we get

()

for any x ∈ U^′ and t ∈ T^′.

For the case of n = 3, the inequality (34) can be rewritten as

()

for all x ∈ U^′ and t ∈ T^′.

Remark 3. It is an open problem whether the wave equation (2) has the generalized Hyers-Ulam stability for the case of either T = (0, t₂) and U = {x ∈ ℝⁿ : 0<|x | < at₂} or T = (t₁, ∞) and U = {x ∈ ℝⁿ:|x | > bt₁} or T = (0, ∞) and U = {x ∈ ℝⁿ:|x | > 0}.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557). This work was also supported by the 2011 Hongik University Research Fund.

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On the Stability of Wave Equation

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1. Introduction

2. Main Results

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On the Stability of Wave Equation

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1. Introduction

2. Main Results

3. Remarks

Conflict of Interests

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